Accelerated Dinkelbach method

TL;DR

This paper introduces an accelerated Dinkelbach method that significantly improves convergence rates for fractional programming by applying a minimal correction, achieving superquadratic and cubic convergence while solving only one subproblem per iteration.

Contribution

It proposes a novel, globally convergent, non-monotone accelerated Dinkelbach algorithm with superquadratic convergence, enhancing efficiency over classical methods.

Findings

Achieves superquadratic and cubic convergence rates.

Maintains practicality by solving only one subproblem per iteration.

Surpasses the quadratic convergence of traditional Dinkelbach methods.

Abstract

The classical Dinkelbach method (1967) solves fractional programming via a parametric approach, generating a decreasing upper bound sequence that converges to the optimum. Its important variant, the interval Dinkelbach method (1991), constructs convergent upper and lower bound sequences that bracket the solution and achieve quadratic and superlinear convergence, respectively, under the assumption that the parametric function is twice continuously differentiable. However, this paper demonstrates that a minimal correction, applied solely to the upper bound iterate, is sufficient to boost the convergence of the method, achieving superquadratic and cubic rates for the upper and lower bound sequences, respectively. By strategically integrating this correction, we develop a globally convergent, non-monotone, and accelerated Dinkelbach algorithm-the first of its kind, to our knowledge. Under sufficient differentiability, the new method achieves an asymptotic average convergence order of at least the square root of 5 per iteration, surpassing the quadratic order of the original algorithm. Crucially, this acceleration is achieved while maintaining the key practicality of solving only a single subproblem per iteration.